Properties

Label 3024.2.b.r.1567.3
Level 3024
Weight 2
Character 3024.1567
Analytic conductor 24.147
Analytic rank 0
Dimension 4
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.r.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410i q^{5} -2.64575 q^{7} +O(q^{10})\) \(q+3.46410i q^{5} -2.64575 q^{7} -4.58258i q^{13} +1.73205i q^{17} -5.29150 q^{19} -4.58258i q^{23} -7.00000 q^{25} +7.93725 q^{29} -2.64575 q^{31} -9.16515i q^{35} +4.00000 q^{37} -6.92820i q^{41} +1.73205i q^{43} -6.00000 q^{47} +7.00000 q^{49} -7.93725 q^{53} -3.00000 q^{59} -9.16515i q^{61} +15.8745 q^{65} -12.1244i q^{67} -4.58258i q^{71} +9.16515i q^{73} +6.92820i q^{79} +12.0000 q^{83} -6.00000 q^{85} -5.19615i q^{89} +12.1244i q^{91} -18.3303i q^{95} +9.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 28q^{25} + 16q^{37} - 24q^{47} + 28q^{49} - 12q^{59} + 48q^{83} - 24q^{85} + O(q^{100}) \)

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
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 4.58258i − 1.27098i −0.772110 0.635489i \(-0.780799\pi\)
0.772110 0.635489i \(-0.219201\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205i 0.420084i 0.977692 + 0.210042i \(0.0673601\pi\)
−0.977692 + 0.210042i \(0.932640\pi\)
\(18\) 0 0
\(19\) −5.29150 −1.21395 −0.606977 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.58258i − 0.955533i −0.878487 0.477767i \(-0.841446\pi\)
0.878487 0.477767i \(-0.158554\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.93725 1.47391 0.736956 0.675941i \(-0.236263\pi\)
0.736956 + 0.675941i \(0.236263\pi\)
\(30\) 0 0
\(31\) −2.64575 −0.475191 −0.237595 0.971364i \(-0.576359\pi\)
−0.237595 + 0.971364i \(0.576359\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 9.16515i − 1.54919i
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.92820i − 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 1.73205i 0.264135i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.93725 −1.09027 −0.545133 0.838350i \(-0.683521\pi\)
−0.545133 + 0.838350i \(0.683521\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) − 9.16515i − 1.17348i −0.809776 0.586739i \(-0.800412\pi\)
0.809776 0.586739i \(-0.199588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.8745 1.96899
\(66\) 0 0
\(67\) − 12.1244i − 1.48123i −0.671932 0.740613i \(-0.734535\pi\)
0.671932 0.740613i \(-0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 4.58258i − 0.543852i −0.962318 0.271926i \(-0.912339\pi\)
0.962318 0.271926i \(-0.0876605\pi\)
\(72\) 0 0
\(73\) 9.16515i 1.07270i 0.843996 + 0.536350i \(0.180197\pi\)
−0.843996 + 0.536350i \(0.819803\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 5.19615i − 0.550791i −0.961331 0.275396i \(-0.911191\pi\)
0.961331 0.275396i \(-0.0888088\pi\)
\(90\) 0 0
\(91\) 12.1244i 1.27098i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 18.3303i − 1.88065i
\(96\) 0 0
\(97\) 9.16515i 0.930580i 0.885158 + 0.465290i \(0.154050\pi\)
−0.885158 + 0.465290i \(0.845950\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410i 0.344691i 0.985037 + 0.172345i \(0.0551346\pi\)
−0.985037 + 0.172345i \(0.944865\pi\)
\(102\) 0 0
\(103\) 13.2288 1.30347 0.651734 0.758448i \(-0.274042\pi\)
0.651734 + 0.758448i \(0.274042\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 18.3303i − 1.77206i −0.463631 0.886029i \(-0.653453\pi\)
0.463631 0.886029i \(-0.346547\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.8745 1.49335 0.746674 0.665190i \(-0.231650\pi\)
0.746674 + 0.665190i \(0.231650\pi\)
\(114\) 0 0
\(115\) 15.8745 1.48031
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 4.58258i − 0.420084i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 6.92820i − 0.619677i
\(126\) 0 0
\(127\) − 13.8564i − 1.22956i −0.788700 0.614779i \(-0.789245\pi\)
0.788700 0.614779i \(-0.210755\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) 14.0000 1.21395
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.8745 1.35625 0.678125 0.734946i \(-0.262793\pi\)
0.678125 + 0.734946i \(0.262793\pi\)
\(138\) 0 0
\(139\) 5.29150 0.448819 0.224410 0.974495i \(-0.427955\pi\)
0.224410 + 0.974495i \(0.427955\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 27.4955i 2.28337i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.93725 −0.650245 −0.325123 0.945672i \(-0.605406\pi\)
−0.325123 + 0.945672i \(0.605406\pi\)
\(150\) 0 0
\(151\) − 3.46410i − 0.281905i −0.990016 0.140952i \(-0.954984\pi\)
0.990016 0.140952i \(-0.0450164\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 9.16515i − 0.736163i
\(156\) 0 0
\(157\) − 22.9129i − 1.82865i −0.404985 0.914323i \(-0.632723\pi\)
0.404985 0.914323i \(-0.367277\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.1244i 0.955533i
\(162\) 0 0
\(163\) 5.19615i 0.406994i 0.979076 + 0.203497i \(0.0652307\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −8.00000 −0.615385
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 17.3205i − 1.31685i −0.752645 0.658427i \(-0.771222\pi\)
0.752645 0.658427i \(-0.228778\pi\)
\(174\) 0 0
\(175\) 18.5203 1.40000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 9.16515i − 0.685036i −0.939511 0.342518i \(-0.888720\pi\)
0.939511 0.342518i \(-0.111280\pi\)
\(180\) 0 0
\(181\) 13.7477i 1.02186i 0.859622 + 0.510930i \(0.170699\pi\)
−0.859622 + 0.510930i \(0.829301\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 13.8564i 1.01874i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −13.2288 −0.937762 −0.468881 0.883261i \(-0.655343\pi\)
−0.468881 + 0.883261i \(0.655343\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −21.0000 −1.47391
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.1244i 0.834675i 0.908752 + 0.417338i \(0.137037\pi\)
−0.908752 + 0.417338i \(0.862963\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.93725 0.533917
\(222\) 0 0
\(223\) −5.29150 −0.354345 −0.177173 0.984180i \(-0.556695\pi\)
−0.177173 + 0.984180i \(0.556695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 0 0
\(229\) − 9.16515i − 0.605650i −0.953046 0.302825i \(-0.902070\pi\)
0.953046 0.302825i \(-0.0979298\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.8745 1.03997 0.519987 0.854174i \(-0.325937\pi\)
0.519987 + 0.854174i \(0.325937\pi\)
\(234\) 0 0
\(235\) − 20.7846i − 1.35584i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 18.3303i − 1.18569i −0.805317 0.592844i \(-0.798005\pi\)
0.805317 0.592844i \(-0.201995\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 24.2487i 1.54919i
\(246\) 0 0
\(247\) 24.2487i 1.54291i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.92820i 0.432169i 0.976375 + 0.216085i \(0.0693287\pi\)
−0.976375 + 0.216085i \(0.930671\pi\)
\(258\) 0 0
\(259\) −10.5830 −0.657596
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 22.9129i − 1.41287i −0.707779 0.706434i \(-0.750303\pi\)
0.707779 0.706434i \(-0.249697\pi\)
\(264\) 0 0
\(265\) − 27.4955i − 1.68903i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.8564i 0.844840i 0.906400 + 0.422420i \(0.138819\pi\)
−0.906400 + 0.422420i \(0.861181\pi\)
\(270\) 0 0
\(271\) −18.5203 −1.12503 −0.562513 0.826789i \(-0.690165\pi\)
−0.562513 + 0.826789i \(0.690165\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −15.8745 −0.946994 −0.473497 0.880795i \(-0.657008\pi\)
−0.473497 + 0.880795i \(0.657008\pi\)
\(282\) 0 0
\(283\) −21.1660 −1.25819 −0.629094 0.777329i \(-0.716574\pi\)
−0.629094 + 0.777329i \(0.716574\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.3303i 1.08200i
\(288\) 0 0
\(289\) 14.0000 0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 31.1769i − 1.82137i −0.413096 0.910687i \(-0.635553\pi\)
0.413096 0.910687i \(-0.364447\pi\)
\(294\) 0 0
\(295\) − 10.3923i − 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −21.0000 −1.21446
\(300\) 0 0
\(301\) − 4.58258i − 0.264135i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 31.7490 1.81794
\(306\) 0 0
\(307\) −21.1660 −1.20801 −0.604004 0.796981i \(-0.706429\pi\)
−0.604004 + 0.796981i \(0.706429\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 27.4955i − 1.55413i −0.629417 0.777067i \(-0.716706\pi\)
0.629417 0.777067i \(-0.283294\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −31.7490 −1.78320 −0.891601 0.452822i \(-0.850417\pi\)
−0.891601 + 0.452822i \(0.850417\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 9.16515i − 0.509963i
\(324\) 0 0
\(325\) 32.0780i 1.77937i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.8745 0.875190
\(330\) 0 0
\(331\) − 29.4449i − 1.61844i −0.587508 0.809218i \(-0.699891\pi\)
0.587508 0.809218i \(-0.300109\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 42.0000 2.29471
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.4955i 1.47603i 0.674782 + 0.738017i \(0.264237\pi\)
−0.674782 + 0.738017i \(0.735763\pi\)
\(348\) 0 0
\(349\) − 13.7477i − 0.735899i −0.929846 0.367949i \(-0.880060\pi\)
0.929846 0.367949i \(-0.119940\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.73205i 0.0921878i 0.998937 + 0.0460939i \(0.0146773\pi\)
−0.998937 + 0.0460939i \(0.985323\pi\)
\(354\) 0 0
\(355\) 15.8745 0.842531
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0780i 1.69301i 0.532378 + 0.846507i \(0.321299\pi\)
−0.532378 + 0.846507i \(0.678701\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −31.7490 −1.66182
\(366\) 0 0
\(367\) 2.64575 0.138107 0.0690535 0.997613i \(-0.478002\pi\)
0.0690535 + 0.997613i \(0.478002\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 21.0000 1.09027
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 36.3731i − 1.87331i
\(378\) 0 0
\(379\) 10.3923i 0.533817i 0.963722 + 0.266908i \(0.0860021\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −31.7490 −1.60974 −0.804869 0.593452i \(-0.797765\pi\)
−0.804869 + 0.593452i \(0.797765\pi\)
\(390\) 0 0
\(391\) 7.93725 0.401404
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) 9.16515i 0.459986i 0.973192 + 0.229993i \(0.0738703\pi\)
−0.973192 + 0.229993i \(0.926130\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.8745 0.792735 0.396368 0.918092i \(-0.370271\pi\)
0.396368 + 0.918092i \(0.370271\pi\)
\(402\) 0 0
\(403\) 12.1244i 0.603957i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 27.4955i − 1.35956i −0.733415 0.679781i \(-0.762075\pi\)
0.733415 0.679781i \(-0.237925\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.93725 0.390567
\(414\) 0 0
\(415\) 41.5692i 2.04055i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 12.1244i − 0.588118i
\(426\) 0 0
\(427\) 24.2487i 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.3303i 0.882940i 0.897276 + 0.441470i \(0.145543\pi\)
−0.897276 + 0.441470i \(0.854457\pi\)
\(432\) 0 0
\(433\) 18.3303i 0.880898i 0.897778 + 0.440449i \(0.145181\pi\)
−0.897778 + 0.440449i \(0.854819\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.2487i 1.15997i
\(438\) 0 0
\(439\) −13.2288 −0.631374 −0.315687 0.948863i \(-0.602235\pi\)
−0.315687 + 0.948863i \(0.602235\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 18.3303i − 0.870899i −0.900213 0.435449i \(-0.856589\pi\)
0.900213 0.435449i \(-0.143411\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.8745 0.749164 0.374582 0.927194i \(-0.377786\pi\)
0.374582 + 0.927194i \(0.377786\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −42.0000 −1.96899
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 31.1769i − 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(462\) 0 0
\(463\) − 38.1051i − 1.77090i −0.464739 0.885448i \(-0.653852\pi\)
0.464739 0.885448i \(-0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 32.0780i 1.48123i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 37.0405 1.69954
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) − 18.3303i − 0.835790i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −31.7490 −1.44165
\(486\) 0 0
\(487\) − 24.2487i − 1.09881i −0.835555 0.549407i \(-0.814854\pi\)
0.835555 0.549407i \(-0.185146\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.6606i 1.65447i 0.561856 + 0.827235i \(0.310087\pi\)
−0.561856 + 0.827235i \(0.689913\pi\)
\(492\) 0 0
\(493\) 13.7477i 0.619166i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.1244i 0.543852i
\(498\) 0 0
\(499\) − 31.1769i − 1.39567i −0.716258 0.697835i \(-0.754147\pi\)
0.716258 0.697835i \(-0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.7128i 1.22835i 0.789170 + 0.614174i \(0.210511\pi\)
−0.789170 + 0.614174i \(0.789489\pi\)
\(510\) 0 0
\(511\) − 24.2487i − 1.07270i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 45.8258i 2.01932i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.9808i 1.13824i 0.822255 + 0.569119i \(0.192716\pi\)
−0.822255 + 0.569119i \(0.807284\pi\)
\(522\) 0 0
\(523\) −10.5830 −0.462763 −0.231381 0.972863i \(-0.574324\pi\)
−0.231381 + 0.972863i \(0.574324\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.58258i − 0.199620i
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −31.7490 −1.37520
\(534\) 0 0
\(535\) 63.4980 2.74526
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 55.4256i − 2.37417i
\(546\) 0 0
\(547\) 38.1051i 1.62926i 0.579983 + 0.814629i \(0.303059\pi\)
−0.579983 + 0.814629i \(0.696941\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) − 18.3303i − 0.779484i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.93725 −0.336312 −0.168156 0.985760i \(-0.553781\pi\)
−0.168156 + 0.985760i \(0.553781\pi\)
\(558\) 0 0
\(559\) 7.93725 0.335710
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) 54.9909i 2.31348i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 15.5885i 0.652357i 0.945308 + 0.326178i \(0.105761\pi\)
−0.945308 + 0.326178i \(0.894239\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 32.0780i 1.33775i
\(576\) 0 0
\(577\) 18.3303i 0.763100i 0.924348 + 0.381550i \(0.124610\pi\)
−0.924348 + 0.381550i \(0.875390\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −31.7490 −1.31717
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 20.7846i − 0.853522i −0.904365 0.426761i \(-0.859655\pi\)
0.904365 0.426761i \(-0.140345\pi\)
\(594\) 0 0
\(595\) 15.8745 0.650791
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0780i 1.31067i 0.755337 + 0.655336i \(0.227473\pi\)
−0.755337 + 0.655336i \(0.772527\pi\)
\(600\) 0 0
\(601\) − 9.16515i − 0.373854i −0.982374 0.186927i \(-0.940147\pi\)
0.982374 0.186927i \(-0.0598528\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 38.1051i 1.54919i
\(606\) 0 0
\(607\) 18.5203 0.751714 0.375857 0.926678i \(-0.377348\pi\)
0.375857 + 0.926678i \(0.377348\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.4955i 1.11235i
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 5.29150 0.212683 0.106342 0.994330i \(-0.466086\pi\)
0.106342 + 0.994330i \(0.466086\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.7477i 0.550791i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.92820i 0.276246i
\(630\) 0 0
\(631\) 10.3923i 0.413711i 0.978371 + 0.206856i \(0.0663230\pi\)
−0.978371 + 0.206856i \(0.933677\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 48.0000 1.90482
\(636\) 0 0
\(637\) − 32.0780i − 1.27098i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.8745 −0.627005 −0.313503 0.949587i \(-0.601502\pi\)
−0.313503 + 0.949587i \(0.601502\pi\)
\(642\) 0 0
\(643\) −5.29150 −0.208676 −0.104338 0.994542i \(-0.533272\pi\)
−0.104338 + 0.994542i \(0.533272\pi\)
\(644\) 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
\(652\) 0 0
\(653\) 23.8118 0.931826 0.465913 0.884830i \(-0.345726\pi\)
0.465913 + 0.884830i \(0.345726\pi\)
\(654\) 0 0
\(655\) 72.7461i 2.84243i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.3303i 0.714047i 0.934095 + 0.357024i \(0.116208\pi\)
−0.934095 + 0.357024i \(0.883792\pi\)
\(660\) 0 0
\(661\) 9.16515i 0.356483i 0.983987 + 0.178242i \(0.0570408\pi\)
−0.983987 + 0.178242i \(0.942959\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 48.4974i 1.88065i
\(666\) 0 0
\(667\) − 36.3731i − 1.40837i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 41.5692i − 1.59763i −0.601574 0.798817i \(-0.705459\pi\)
0.601574 0.798817i \(-0.294541\pi\)
\(678\) 0 0
\(679\) − 24.2487i − 0.930580i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.16515i − 0.350695i −0.984507 0.175347i \(-0.943895\pi\)
0.984507 0.175347i \(-0.0561049\pi\)
\(684\) 0 0
\(685\) 54.9909i 2.10109i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 36.3731i 1.38570i
\(690\) 0 0
\(691\) 21.1660 0.805193 0.402596 0.915378i \(-0.368108\pi\)
0.402596 + 0.915378i \(0.368108\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.3303i 0.695308i
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.7490 −1.19914 −0.599572 0.800321i \(-0.704662\pi\)
−0.599572 + 0.800321i \(0.704662\pi\)
\(702\) 0 0
\(703\) −21.1660 −0.798291
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.16515i − 0.344691i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.1244i 0.454061i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −35.0000 −1.30347
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −55.5608 −2.06348
\(726\) 0 0
\(727\) −34.3948 −1.27563 −0.637816 0.770189i \(-0.720162\pi\)
−0.637816 + 0.770189i \(0.720162\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) − 22.9129i − 0.846306i −0.906058 0.423153i \(-0.860923\pi\)
0.906058 0.423153i \(-0.139077\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 24.2487i − 0.892003i −0.895032 0.446002i \(-0.852848\pi\)
0.895032 0.446002i \(-0.147152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.7477i 0.504355i 0.967681 + 0.252178i \(0.0811467\pi\)
−0.967681 + 0.252178i \(0.918853\pi\)
\(744\) 0 0
\(745\) − 27.4955i − 1.00736i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 48.4974i 1.77206i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.66025i 0.313934i 0.987604 + 0.156967i \(0.0501716\pi\)
−0.987604 + 0.156967i \(0.949828\pi\)
\(762\) 0 0
\(763\) 42.3320 1.53252
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.7477i 0.496402i
\(768\) 0 0
\(769\) 9.16515i 0.330504i 0.986251 + 0.165252i \(0.0528437\pi\)
−0.986251 + 0.165252i \(0.947156\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 13.8564i − 0.498380i −0.968455 0.249190i \(-0.919836\pi\)
0.968455 0.249190i \(-0.0801644\pi\)
\(774\) 0 0
\(775\) 18.5203 0.665267
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.6606i 1.31350i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 79.3725 2.83293
\(786\) 0 0
\(787\) −26.4575 −0.943108 −0.471554 0.881837i \(-0.656307\pi\)
−0.471554 + 0.881837i \(0.656307\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −42.0000 −1.49335
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34.6410i − 1.22705i −0.789676 0.613524i \(-0.789751\pi\)
0.789676 0.613524i \(-0.210249\pi\)
\(798\) 0 0
\(799\) − 10.3923i − 0.367653i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −42.0000 −1.48031
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −47.6235 −1.67435 −0.837177 0.546932i \(-0.815796\pi\)
−0.837177 + 0.546932i \(0.815796\pi\)
\(810\) 0 0
\(811\) −42.3320 −1.48648 −0.743239 0.669026i \(-0.766712\pi\)
−0.743239 + 0.669026i \(0.766712\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) − 9.16515i − 0.320648i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.93725 −0.277012 −0.138506 0.990362i \(-0.544230\pi\)
−0.138506 + 0.990362i \(0.544230\pi\)
\(822\) 0 0
\(823\) − 3.46410i − 0.120751i −0.998176 0.0603755i \(-0.980770\pi\)
0.998176 0.0603755i \(-0.0192298\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.4955i 0.956111i 0.878330 + 0.478055i \(0.158658\pi\)
−0.878330 + 0.478055i \(0.841342\pi\)
\(828\) 0 0
\(829\) − 45.8258i − 1.59159i −0.605563 0.795797i \(-0.707052\pi\)
0.605563 0.795797i \(-0.292948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.1244i 0.420084i
\(834\) 0 0
\(835\) − 41.5692i − 1.43856i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 34.0000 1.17241
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 27.7128i − 0.953350i
\(846\) 0 0
\(847\) −29.1033 −1.00000
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 18.3303i − 0.628355i
\(852\) 0 0
\(853\) − 32.0780i − 1.09833i −0.835714 0.549165i \(-0.814946\pi\)
0.835714 0.549165i \(-0.185054\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.73205i − 0.0591657i −0.999562 0.0295829i \(-0.990582\pi\)
0.999562 0.0295829i \(-0.00941789\pi\)
\(858\) 0 0
\(859\) −52.9150 −1.80544 −0.902719 0.430231i \(-0.858432\pi\)
−0.902719 + 0.430231i \(0.858432\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 41.2432i − 1.40393i −0.712209 0.701967i \(-0.752305\pi\)
0.712209 0.701967i \(-0.247695\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −55.5608 −1.88261
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.3303i 0.619677i
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 50.2295i − 1.69227i −0.532965 0.846137i \(-0.678922\pi\)
0.532965 0.846137i \(-0.321078\pi\)
\(882\) 0 0
\(883\) 22.5167i 0.757746i 0.925449 + 0.378873i \(0.123688\pi\)
−0.925449 + 0.378873i \(0.876312\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 36.6606i 1.22956i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 31.7490 1.06244
\(894\) 0 0
\(895\) 31.7490 1.06125
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) − 13.7477i − 0.458003i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −47.6235 −1.58306
\(906\) 0 0
\(907\) 45.0333i 1.49531i 0.664089 + 0.747653i \(0.268820\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 54.9909i 1.82193i 0.412484 + 0.910965i \(0.364661\pi\)
−0.412484 + 0.910965i \(0.635339\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −55.5608 −1.83478
\(918\) 0 0
\(919\) 41.5692i 1.37124i 0.727959 + 0.685621i \(0.240469\pi\)
−0.727959 + 0.685621i \(0.759531\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −21.0000 −0.691223
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 27.7128i − 0.909228i −0.890689 0.454614i \(-0.849777\pi\)
0.890689 0.454614i \(-0.150223\pi\)
\(930\) 0 0
\(931\) −37.0405 −1.21395
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 27.4955i − 0.898237i −0.893472 0.449119i \(-0.851738\pi\)
0.893472 0.449119i \(-0.148262\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 10.3923i − 0.338779i −0.985549 0.169390i \(-0.945820\pi\)
0.985549 0.169390i \(-0.0541797\pi\)
\(942\) 0 0
\(943\) −31.7490 −1.03389
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.8258i 1.48914i 0.667546 + 0.744569i \(0.267345\pi\)
−0.667546 + 0.744569i \(0.732655\pi\)
\(948\) 0 0
\(949\) 42.0000 1.36338
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −42.0000 −1.35625
\(960\) 0 0
\(961\) −24.0000 −0.774194
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 79.6743i − 2.56481i
\(966\) 0 0
\(967\) 24.2487i 0.779786i 0.920860 + 0.389893i \(0.127488\pi\)
−0.920860 + 0.389893i \(0.872512\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.6235 1.52361 0.761806 0.647806i \(-0.224313\pi\)
0.761806 + 0.647806i \(0.224313\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.93725 0.252390
\(990\) 0 0
\(991\) 3.46410i 0.110041i 0.998485 + 0.0550204i \(0.0175224\pi\)
−0.998485 + 0.0550204i \(0.982478\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 45.8258i − 1.45277i
\(996\) 0 0
\(997\) − 13.7477i − 0.435395i −0.976016 0.217697i \(-0.930145\pi\)
0.976016 0.217697i \(-0.0698546\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3024.2.b.r.1567.3 yes 4
3.2 odd 2 3024.2.b.s.1567.1 yes 4
4.3 odd 2 3024.2.b.s.1567.4 yes 4
7.6 odd 2 3024.2.b.s.1567.2 yes 4
12.11 even 2 inner 3024.2.b.r.1567.2 yes 4
21.20 even 2 inner 3024.2.b.r.1567.4 yes 4
28.27 even 2 inner 3024.2.b.r.1567.1 4
84.83 odd 2 3024.2.b.s.1567.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.r.1567.1 4 28.27 even 2 inner
3024.2.b.r.1567.2 yes 4 12.11 even 2 inner
3024.2.b.r.1567.3 yes 4 1.1 even 1 trivial
3024.2.b.r.1567.4 yes 4 21.20 even 2 inner
3024.2.b.s.1567.1 yes 4 3.2 odd 2
3024.2.b.s.1567.2 yes 4 7.6 odd 2
3024.2.b.s.1567.3 yes 4 84.83 odd 2
3024.2.b.s.1567.4 yes 4 4.3 odd 2