Properties

Label 6048.2.h.e.2591.6
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.6
Root \(0.965926 - 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.e.2591.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+2.82843i q^{5} -1.00000i q^{7} +6.29253 q^{11} +5.44949 q^{13} +0.317837i q^{17} +4.00000i q^{19} +0.317837 q^{23} -3.00000 q^{25} -5.19615i q^{29} +8.79796i q^{31} +2.82843 q^{35} +0.898979 q^{37} -3.46410i q^{41} -5.44949i q^{43} +12.5851 q^{47} -1.00000 q^{49} -8.02458i q^{53} +17.7980i q^{55} +5.83183 q^{59} +12.8990 q^{61} +15.4135i q^{65} +3.44949i q^{67} -2.51059 q^{71} -4.00000 q^{73} -6.29253i q^{77} -10.0000i q^{79} -3.46410 q^{83} -0.898979 q^{85} -10.7101i q^{89} -5.44949i q^{91} -11.3137 q^{95} -15.7980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 24q^{13} - 24q^{25} - 32q^{37} - 8q^{49} + 64q^{61} - 32q^{73} + 32q^{85} - 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) 2.82843i 1.26491i 0.774597 + 0.632456i \(0.217953\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.29253 1.89727 0.948634 0.316374i \(-0.102466\pi\)
0.948634 + 0.316374i \(0.102466\pi\)
\(12\) 0 0
\(13\) 5.44949 1.51142 0.755708 0.654908i \(-0.227293\pi\)
0.755708 + 0.654908i \(0.227293\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.317837i 0.0770869i 0.999257 + 0.0385434i \(0.0122718\pi\)
−0.999257 + 0.0385434i \(0.987728\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.317837 0.0662736 0.0331368 0.999451i \(-0.489450\pi\)
0.0331368 + 0.999451i \(0.489450\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.19615i − 0.964901i −0.875923 0.482451i \(-0.839747\pi\)
0.875923 0.482451i \(-0.160253\pi\)
\(30\) 0 0
\(31\) 8.79796i 1.58016i 0.613004 + 0.790080i \(0.289961\pi\)
−0.613004 + 0.790080i \(0.710039\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) 0.898979 0.147791 0.0738957 0.997266i \(-0.476457\pi\)
0.0738957 + 0.997266i \(0.476457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) − 5.44949i − 0.831039i −0.909584 0.415520i \(-0.863600\pi\)
0.909584 0.415520i \(-0.136400\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.5851 1.83572 0.917860 0.396905i \(-0.129916\pi\)
0.917860 + 0.396905i \(0.129916\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.02458i − 1.10226i −0.834419 0.551130i \(-0.814197\pi\)
0.834419 0.551130i \(-0.185803\pi\)
\(54\) 0 0
\(55\) 17.7980i 2.39988i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.83183 0.759239 0.379620 0.925143i \(-0.376055\pi\)
0.379620 + 0.925143i \(0.376055\pi\)
\(60\) 0 0
\(61\) 12.8990 1.65155 0.825773 0.564003i \(-0.190739\pi\)
0.825773 + 0.564003i \(0.190739\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.4135i 1.91181i
\(66\) 0 0
\(67\) 3.44949i 0.421422i 0.977548 + 0.210711i \(0.0675779\pi\)
−0.977548 + 0.210711i \(0.932422\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.51059 −0.297952 −0.148976 0.988841i \(-0.547598\pi\)
−0.148976 + 0.988841i \(0.547598\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.29253i − 0.717100i
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 0.562544i \(-0.809823\pi\)
0.826767 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) −0.898979 −0.0975080
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 10.7101i − 1.13527i −0.823279 0.567636i \(-0.807858\pi\)
0.823279 0.567636i \(-0.192142\pi\)
\(90\) 0 0
\(91\) − 5.44949i − 0.571262i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −11.3137 −1.16076
\(96\) 0 0
\(97\) −15.7980 −1.60404 −0.802020 0.597297i \(-0.796241\pi\)
−0.802020 + 0.597297i \(0.796241\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.65685i − 0.562878i −0.959579 0.281439i \(-0.909188\pi\)
0.959579 0.281439i \(-0.0908117\pi\)
\(102\) 0 0
\(103\) 3.89898i 0.384178i 0.981378 + 0.192089i \(0.0615262\pi\)
−0.981378 + 0.192089i \(0.938474\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.2419 −1.76351 −0.881756 0.471706i \(-0.843638\pi\)
−0.881756 + 0.471706i \(0.843638\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 8.48528i − 0.798228i −0.916901 0.399114i \(-0.869318\pi\)
0.916901 0.399114i \(-0.130682\pi\)
\(114\) 0 0
\(115\) 0.898979i 0.0838303i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.317837 0.0291361
\(120\) 0 0
\(121\) 28.5959 2.59963
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 10.8990i 0.967128i 0.875309 + 0.483564i \(0.160658\pi\)
−0.875309 + 0.483564i \(0.839342\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.36773 −0.206869 −0.103435 0.994636i \(-0.532983\pi\)
−0.103435 + 0.994636i \(0.532983\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 19.5133i − 1.66713i −0.552421 0.833565i \(-0.686296\pi\)
0.552421 0.833565i \(-0.313704\pi\)
\(138\) 0 0
\(139\) − 15.7980i − 1.33997i −0.742377 0.669983i \(-0.766302\pi\)
0.742377 0.669983i \(-0.233698\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 34.2911 2.86756
\(144\) 0 0
\(145\) 14.6969 1.22051
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.02458i 0.657399i 0.944435 + 0.328700i \(0.106610\pi\)
−0.944435 + 0.328700i \(0.893390\pi\)
\(150\) 0 0
\(151\) 5.79796i 0.471831i 0.971774 + 0.235916i \(0.0758089\pi\)
−0.971774 + 0.235916i \(0.924191\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −24.8844 −1.99876
\(156\) 0 0
\(157\) 16.1464 1.28863 0.644313 0.764762i \(-0.277144\pi\)
0.644313 + 0.764762i \(0.277144\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 0.317837i − 0.0250491i
\(162\) 0 0
\(163\) 2.55051i 0.199771i 0.994999 + 0.0998857i \(0.0318477\pi\)
−0.994999 + 0.0998857i \(0.968152\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0492 −1.24192 −0.620961 0.783842i \(-0.713257\pi\)
−0.620961 + 0.783842i \(0.713257\pi\)
\(168\) 0 0
\(169\) 16.6969 1.28438
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 0.921404i − 0.0700530i −0.999386 0.0350265i \(-0.988848\pi\)
0.999386 0.0350265i \(-0.0111516\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.0280 −0.824270 −0.412135 0.911123i \(-0.635217\pi\)
−0.412135 + 0.911123i \(0.635217\pi\)
\(180\) 0 0
\(181\) −9.65153 −0.717393 −0.358696 0.933454i \(-0.616779\pi\)
−0.358696 + 0.933454i \(0.616779\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.54270i 0.186943i
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.0280 0.797957 0.398978 0.916960i \(-0.369365\pi\)
0.398978 + 0.916960i \(0.369365\pi\)
\(192\) 0 0
\(193\) −3.89898 −0.280655 −0.140327 0.990105i \(-0.544815\pi\)
−0.140327 + 0.990105i \(0.544815\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.9706i 1.20910i 0.796566 + 0.604551i \(0.206648\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(198\) 0 0
\(199\) 20.7980i 1.47433i 0.675714 + 0.737164i \(0.263836\pi\)
−0.675714 + 0.737164i \(0.736164\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.19615 −0.364698
\(204\) 0 0
\(205\) 9.79796 0.684319
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 25.1701i 1.74105i
\(210\) 0 0
\(211\) 14.1464i 0.973880i 0.873435 + 0.486940i \(0.161887\pi\)
−0.873435 + 0.486940i \(0.838113\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.4135 1.05119
\(216\) 0 0
\(217\) 8.79796 0.597244
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.73205i 0.116510i
\(222\) 0 0
\(223\) 9.79796i 0.656120i 0.944657 + 0.328060i \(0.106395\pi\)
−0.944657 + 0.328060i \(0.893605\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.460702 −0.0305779 −0.0152889 0.999883i \(-0.504867\pi\)
−0.0152889 + 0.999883i \(0.504867\pi\)
\(228\) 0 0
\(229\) −3.10102 −0.204921 −0.102461 0.994737i \(-0.532672\pi\)
−0.102461 + 0.994737i \(0.532672\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 27.0771i 1.77388i 0.461882 + 0.886941i \(0.347174\pi\)
−0.461882 + 0.886941i \(0.652826\pi\)
\(234\) 0 0
\(235\) 35.5959i 2.32202i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.82843 −0.182956 −0.0914779 0.995807i \(-0.529159\pi\)
−0.0914779 + 0.995807i \(0.529159\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.82843i − 0.180702i
\(246\) 0 0
\(247\) 21.7980i 1.38697i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.73545 −0.298899 −0.149449 0.988769i \(-0.547750\pi\)
−0.149449 + 0.988769i \(0.547750\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 26.0915i − 1.62754i −0.581184 0.813772i \(-0.697410\pi\)
0.581184 0.813772i \(-0.302590\pi\)
\(258\) 0 0
\(259\) − 0.898979i − 0.0558599i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.6527 1.02685 0.513426 0.858134i \(-0.328376\pi\)
0.513426 + 0.858134i \(0.328376\pi\)
\(264\) 0 0
\(265\) 22.6969 1.39426
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.4921i 0.883598i 0.897114 + 0.441799i \(0.145660\pi\)
−0.897114 + 0.441799i \(0.854340\pi\)
\(270\) 0 0
\(271\) − 29.8990i − 1.81623i −0.418717 0.908117i \(-0.637520\pi\)
0.418717 0.908117i \(-0.362480\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.8776 −1.13836
\(276\) 0 0
\(277\) 30.8990 1.85654 0.928270 0.371907i \(-0.121296\pi\)
0.928270 + 0.371907i \(0.121296\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.57826i 0.392426i 0.980561 + 0.196213i \(0.0628644\pi\)
−0.980561 + 0.196213i \(0.937136\pi\)
\(282\) 0 0
\(283\) 5.79796i 0.344653i 0.985040 + 0.172326i \(0.0551284\pi\)
−0.985040 + 0.172326i \(0.944872\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.46410 −0.204479
\(288\) 0 0
\(289\) 16.8990 0.994058
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 13.2207i − 0.772363i −0.922423 0.386182i \(-0.873794\pi\)
0.922423 0.386182i \(-0.126206\pi\)
\(294\) 0 0
\(295\) 16.4949i 0.960370i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.73205 0.100167
\(300\) 0 0
\(301\) −5.44949 −0.314103
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 36.4838i 2.08906i
\(306\) 0 0
\(307\) 26.6969i 1.52367i 0.647768 + 0.761837i \(0.275702\pi\)
−0.647768 + 0.761837i \(0.724298\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.4203 1.21463 0.607316 0.794460i \(-0.292246\pi\)
0.607316 + 0.794460i \(0.292246\pi\)
\(312\) 0 0
\(313\) 15.5959 0.881533 0.440767 0.897622i \(-0.354707\pi\)
0.440767 + 0.897622i \(0.354707\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.5851i 0.706847i 0.935463 + 0.353424i \(0.114983\pi\)
−0.935463 + 0.353424i \(0.885017\pi\)
\(318\) 0 0
\(319\) − 32.6969i − 1.83068i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.27135 −0.0707397
\(324\) 0 0
\(325\) −16.3485 −0.906850
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 12.5851i − 0.693837i
\(330\) 0 0
\(331\) − 5.65153i − 0.310636i −0.987865 0.155318i \(-0.950360\pi\)
0.987865 0.155318i \(-0.0496403\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.75663 −0.533062
\(336\) 0 0
\(337\) −19.8990 −1.08397 −0.541983 0.840389i \(-0.682326\pi\)
−0.541983 + 0.840389i \(0.682326\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 55.3614i 2.99799i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.1489 −1.08165 −0.540826 0.841135i \(-0.681888\pi\)
−0.540826 + 0.841135i \(0.681888\pi\)
\(348\) 0 0
\(349\) 28.8434 1.54395 0.771975 0.635653i \(-0.219269\pi\)
0.771975 + 0.635653i \(0.219269\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.7019i 1.74055i 0.492570 + 0.870273i \(0.336058\pi\)
−0.492570 + 0.870273i \(0.663942\pi\)
\(354\) 0 0
\(355\) − 7.10102i − 0.376883i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.61037 0.348882 0.174441 0.984668i \(-0.444188\pi\)
0.174441 + 0.984668i \(0.444188\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 11.3137i − 0.592187i
\(366\) 0 0
\(367\) 34.7980i 1.81644i 0.418494 + 0.908219i \(0.362558\pi\)
−0.418494 + 0.908219i \(0.637442\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.02458 −0.416615
\(372\) 0 0
\(373\) −32.2929 −1.67206 −0.836030 0.548683i \(-0.815129\pi\)
−0.836030 + 0.548683i \(0.815129\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 28.3164i − 1.45837i
\(378\) 0 0
\(379\) − 10.0000i − 0.513665i −0.966456 0.256833i \(-0.917321\pi\)
0.966456 0.256833i \(-0.0826790\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.8920 −0.914237 −0.457118 0.889406i \(-0.651118\pi\)
−0.457118 + 0.889406i \(0.651118\pi\)
\(384\) 0 0
\(385\) 17.7980 0.907068
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 36.4838i − 1.84980i −0.380206 0.924902i \(-0.624147\pi\)
0.380206 0.924902i \(-0.375853\pi\)
\(390\) 0 0
\(391\) 0.101021i 0.00510883i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 28.2843 1.42314
\(396\) 0 0
\(397\) −15.1010 −0.757898 −0.378949 0.925417i \(-0.623715\pi\)
−0.378949 + 0.925417i \(0.623715\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.6637i 0.582455i 0.956654 + 0.291228i \(0.0940637\pi\)
−0.956654 + 0.291228i \(0.905936\pi\)
\(402\) 0 0
\(403\) 47.9444i 2.38828i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.65685 0.280400
\(408\) 0 0
\(409\) −28.6969 −1.41897 −0.709486 0.704719i \(-0.751073\pi\)
−0.709486 + 0.704719i \(0.751073\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5.83183i − 0.286965i
\(414\) 0 0
\(415\) − 9.79796i − 0.480963i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.6520 −1.49745 −0.748724 0.662882i \(-0.769333\pi\)
−0.748724 + 0.662882i \(0.769333\pi\)
\(420\) 0 0
\(421\) −31.5959 −1.53989 −0.769945 0.638110i \(-0.779717\pi\)
−0.769945 + 0.638110i \(0.779717\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 0.953512i − 0.0462521i
\(426\) 0 0
\(427\) − 12.8990i − 0.624225i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 26.7272 1.28740 0.643702 0.765276i \(-0.277398\pi\)
0.643702 + 0.765276i \(0.277398\pi\)
\(432\) 0 0
\(433\) −19.5959 −0.941720 −0.470860 0.882208i \(-0.656056\pi\)
−0.470860 + 0.882208i \(0.656056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.27135i 0.0608169i
\(438\) 0 0
\(439\) − 11.4949i − 0.548622i −0.961641 0.274311i \(-0.911550\pi\)
0.961641 0.274311i \(-0.0884497\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.285729 0.0135754 0.00678770 0.999977i \(-0.497839\pi\)
0.00678770 + 0.999977i \(0.497839\pi\)
\(444\) 0 0
\(445\) 30.2929 1.43602
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 29.2699i − 1.38133i −0.723174 0.690666i \(-0.757318\pi\)
0.723174 0.690666i \(-0.242682\pi\)
\(450\) 0 0
\(451\) − 21.7980i − 1.02643i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.4135 0.722595
\(456\) 0 0
\(457\) −10.3031 −0.481957 −0.240978 0.970530i \(-0.577468\pi\)
−0.240978 + 0.970530i \(0.577468\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 6.92820i − 0.322679i −0.986899 0.161339i \(-0.948419\pi\)
0.986899 0.161339i \(-0.0515813\pi\)
\(462\) 0 0
\(463\) 31.5959i 1.46839i 0.678940 + 0.734193i \(0.262439\pi\)
−0.678940 + 0.734193i \(0.737561\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.84961 0.363236 0.181618 0.983369i \(-0.441866\pi\)
0.181618 + 0.983369i \(0.441866\pi\)
\(468\) 0 0
\(469\) 3.44949 0.159283
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 34.2911i − 1.57671i
\(474\) 0 0
\(475\) − 12.0000i − 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.94258 −0.271524 −0.135762 0.990742i \(-0.543348\pi\)
−0.135762 + 0.990742i \(0.543348\pi\)
\(480\) 0 0
\(481\) 4.89898 0.223374
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 44.6834i − 2.02897i
\(486\) 0 0
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.5344 −1.10722 −0.553612 0.832775i \(-0.686751\pi\)
−0.553612 + 0.832775i \(0.686751\pi\)
\(492\) 0 0
\(493\) 1.65153 0.0743812
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.51059i 0.112615i
\(498\) 0 0
\(499\) 35.7980i 1.60254i 0.598305 + 0.801268i \(0.295841\pi\)
−0.598305 + 0.801268i \(0.704159\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.37113 0.239487 0.119743 0.992805i \(-0.461793\pi\)
0.119743 + 0.992805i \(0.461793\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 0.285729i − 0.0126647i −0.999980 0.00633236i \(-0.997984\pi\)
0.999980 0.00633236i \(-0.00201567\pi\)
\(510\) 0 0
\(511\) 4.00000i 0.176950i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11.0280 −0.485951
\(516\) 0 0
\(517\) 79.1918 3.48285
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.6315i 0.509587i 0.966995 + 0.254794i \(0.0820075\pi\)
−0.966995 + 0.254794i \(0.917992\pi\)
\(522\) 0 0
\(523\) 19.7980i 0.865704i 0.901465 + 0.432852i \(0.142493\pi\)
−0.901465 + 0.432852i \(0.857507\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.79632 −0.121810
\(528\) 0 0
\(529\) −22.8990 −0.995608
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 18.8776i − 0.817679i
\(534\) 0 0
\(535\) − 51.5959i − 2.23069i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.29253 −0.271038
\(540\) 0 0
\(541\) 40.4949 1.74101 0.870506 0.492158i \(-0.163792\pi\)
0.870506 + 0.492158i \(0.163792\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 22.6274i − 0.969252i
\(546\) 0 0
\(547\) 15.3939i 0.658195i 0.944296 + 0.329097i \(0.106744\pi\)
−0.944296 + 0.329097i \(0.893256\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.746431i 0.0316273i 0.999875 + 0.0158136i \(0.00503385\pi\)
−0.999875 + 0.0158136i \(0.994966\pi\)
\(558\) 0 0
\(559\) − 29.6969i − 1.25605i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.9951 −1.05342 −0.526710 0.850045i \(-0.676575\pi\)
−0.526710 + 0.850045i \(0.676575\pi\)
\(564\) 0 0
\(565\) 24.0000 1.00969
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.19275i 0.0919250i 0.998943 + 0.0459625i \(0.0146355\pi\)
−0.998943 + 0.0459625i \(0.985365\pi\)
\(570\) 0 0
\(571\) 11.2474i 0.470691i 0.971912 + 0.235346i \(0.0756222\pi\)
−0.971912 + 0.235346i \(0.924378\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.953512 −0.0397642
\(576\) 0 0
\(577\) 5.30306 0.220769 0.110385 0.993889i \(-0.464792\pi\)
0.110385 + 0.993889i \(0.464792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.46410i 0.143715i
\(582\) 0 0
\(583\) − 50.4949i − 2.09128i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.2385 0.628961 0.314480 0.949264i \(-0.398170\pi\)
0.314480 + 0.949264i \(0.398170\pi\)
\(588\) 0 0
\(589\) −35.1918 −1.45005
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 26.0915i − 1.07145i −0.844392 0.535725i \(-0.820038\pi\)
0.844392 0.535725i \(-0.179962\pi\)
\(594\) 0 0
\(595\) 0.898979i 0.0368546i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.6513 −1.82440 −0.912201 0.409744i \(-0.865618\pi\)
−0.912201 + 0.409744i \(0.865618\pi\)
\(600\) 0 0
\(601\) −16.8990 −0.689324 −0.344662 0.938727i \(-0.612006\pi\)
−0.344662 + 0.938727i \(0.612006\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 80.8815i 3.28830i
\(606\) 0 0
\(607\) 19.0000i 0.771186i 0.922669 + 0.385593i \(0.126003\pi\)
−0.922669 + 0.385593i \(0.873997\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 68.5821 2.77454
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 29.2699i − 1.17836i −0.808001 0.589181i \(-0.799451\pi\)
0.808001 0.589181i \(-0.200549\pi\)
\(618\) 0 0
\(619\) 29.5959i 1.18956i 0.803888 + 0.594780i \(0.202761\pi\)
−0.803888 + 0.594780i \(0.797239\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.7101 −0.429093
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.285729i 0.0113928i
\(630\) 0 0
\(631\) 36.4949i 1.45284i 0.687252 + 0.726419i \(0.258817\pi\)
−0.687252 + 0.726419i \(0.741183\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −30.8270 −1.22333
\(636\) 0 0
\(637\) −5.44949 −0.215917
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 20.1489i − 0.795835i −0.917421 0.397918i \(-0.869733\pi\)
0.917421 0.397918i \(-0.130267\pi\)
\(642\) 0 0
\(643\) − 48.4949i − 1.91245i −0.292629 0.956226i \(-0.594530\pi\)
0.292629 0.956226i \(-0.405470\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.94258 −0.233627 −0.116814 0.993154i \(-0.537268\pi\)
−0.116814 + 0.993154i \(0.537268\pi\)
\(648\) 0 0
\(649\) 36.6969 1.44048
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 25.3451i − 0.991830i −0.868371 0.495915i \(-0.834833\pi\)
0.868371 0.495915i \(-0.165167\pi\)
\(654\) 0 0
\(655\) − 6.69694i − 0.261671i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.7846 0.809653 0.404827 0.914393i \(-0.367332\pi\)
0.404827 + 0.914393i \(0.367332\pi\)
\(660\) 0 0
\(661\) 12.8990 0.501712 0.250856 0.968024i \(-0.419288\pi\)
0.250856 + 0.968024i \(0.419288\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.3137i 0.438727i
\(666\) 0 0
\(667\) − 1.65153i − 0.0639475i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 81.1672 3.13342
\(672\) 0 0
\(673\) −28.5959 −1.10229 −0.551146 0.834409i \(-0.685809\pi\)
−0.551146 + 0.834409i \(0.685809\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 28.3485i − 1.08952i −0.838592 0.544760i \(-0.816621\pi\)
0.838592 0.544760i \(-0.183379\pi\)
\(678\) 0 0
\(679\) 15.7980i 0.606270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −38.0409 −1.45559 −0.727797 0.685792i \(-0.759456\pi\)
−0.727797 + 0.685792i \(0.759456\pi\)
\(684\) 0 0
\(685\) 55.1918 2.10877
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 43.7299i − 1.66598i
\(690\) 0 0
\(691\) 28.2020i 1.07286i 0.843946 + 0.536428i \(0.180227\pi\)
−0.843946 + 0.536428i \(0.819773\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 44.6834 1.69494
\(696\) 0 0
\(697\) 1.10102 0.0417041
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.65685i 0.213656i 0.994277 + 0.106828i \(0.0340695\pi\)
−0.994277 + 0.106828i \(0.965931\pi\)
\(702\) 0 0
\(703\) 3.59592i 0.135623i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.65685 −0.212748
\(708\) 0 0
\(709\) −41.5959 −1.56217 −0.781084 0.624426i \(-0.785333\pi\)
−0.781084 + 0.624426i \(0.785333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.79632i 0.104723i
\(714\) 0 0
\(715\) 96.9898i 3.62721i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.3349 0.609189 0.304594 0.952482i \(-0.401479\pi\)
0.304594 + 0.952482i \(0.401479\pi\)
\(720\) 0 0
\(721\) 3.89898 0.145206
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.5885i 0.578941i
\(726\) 0 0
\(727\) − 9.49490i − 0.352146i −0.984377 0.176073i \(-0.943660\pi\)
0.984377 0.176073i \(-0.0563395\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.73205 0.0640622
\(732\) 0 0
\(733\) 0.348469 0.0128710 0.00643550 0.999979i \(-0.497952\pi\)
0.00643550 + 0.999979i \(0.497952\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.7060i 0.799551i
\(738\) 0 0
\(739\) − 43.7980i − 1.61113i −0.592505 0.805567i \(-0.701861\pi\)
0.592505 0.805567i \(-0.298139\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.3376 1.22304 0.611518 0.791230i \(-0.290559\pi\)
0.611518 + 0.791230i \(0.290559\pi\)
\(744\) 0 0
\(745\) −22.6969 −0.831551
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.2419i 0.666545i
\(750\) 0 0
\(751\) 44.8990i 1.63839i 0.573517 + 0.819194i \(0.305579\pi\)
−0.573517 + 0.819194i \(0.694421\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.3991 −0.596825
\(756\) 0 0
\(757\) −33.1918 −1.20638 −0.603189 0.797598i \(-0.706103\pi\)
−0.603189 + 0.797598i \(0.706103\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 48.1154i 1.74418i 0.489345 + 0.872090i \(0.337236\pi\)
−0.489345 + 0.872090i \(0.662764\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 31.7805 1.14753
\(768\) 0 0
\(769\) −4.89898 −0.176662 −0.0883309 0.996091i \(-0.528153\pi\)
−0.0883309 + 0.996091i \(0.528153\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 31.4626i − 1.13163i −0.824531 0.565816i \(-0.808561\pi\)
0.824531 0.565816i \(-0.191439\pi\)
\(774\) 0 0
\(775\) − 26.3939i − 0.948096i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) −15.7980 −0.565295
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 45.6690i 1.63000i
\(786\) 0 0
\(787\) − 21.5959i − 0.769811i −0.922956 0.384906i \(-0.874234\pi\)
0.922956 0.384906i \(-0.125766\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.48528 −0.301702
\(792\) 0 0
\(793\) 70.2929 2.49617
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 25.4558i − 0.901692i −0.892602 0.450846i \(-0.851122\pi\)
0.892602 0.450846i \(-0.148878\pi\)
\(798\) 0 0
\(799\) 4.00000i 0.141510i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −25.1701 −0.888234
\(804\) 0 0
\(805\) 0.898979 0.0316849
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.6486i 0.972073i 0.873939 + 0.486036i \(0.161558\pi\)
−0.873939 + 0.486036i \(0.838442\pi\)
\(810\) 0 0
\(811\) − 24.4949i − 0.860132i −0.902797 0.430066i \(-0.858490\pi\)
0.902797 0.430066i \(-0.141510\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.21393 −0.252693
\(816\) 0 0
\(817\) 21.7980 0.762614
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 37.0087i − 1.29161i −0.763501 0.645807i \(-0.776521\pi\)
0.763501 0.645807i \(-0.223479\pi\)
\(822\) 0 0
\(823\) − 21.1010i − 0.735535i −0.929918 0.367768i \(-0.880122\pi\)
0.929918 0.367768i \(-0.119878\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −37.7552 −1.31288 −0.656438 0.754380i \(-0.727938\pi\)
−0.656438 + 0.754380i \(0.727938\pi\)
\(828\) 0 0
\(829\) −46.6969 −1.62185 −0.810926 0.585149i \(-0.801036\pi\)
−0.810926 + 0.585149i \(0.801036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.317837i − 0.0110124i
\(834\) 0 0
\(835\) − 45.3939i − 1.57092i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.46410 0.119594 0.0597970 0.998211i \(-0.480955\pi\)
0.0597970 + 0.998211i \(0.480955\pi\)
\(840\) 0 0
\(841\) 2.00000 0.0689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 47.2261i 1.62463i
\(846\) 0 0
\(847\) − 28.5959i − 0.982567i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.285729 0.00979467
\(852\) 0 0
\(853\) −31.6515 −1.08373 −0.541864 0.840466i \(-0.682281\pi\)
−0.541864 + 0.840466i \(0.682281\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 10.7101i − 0.365851i −0.983127 0.182926i \(-0.941443\pi\)
0.983127 0.182926i \(-0.0585568\pi\)
\(858\) 0 0
\(859\) − 42.2929i − 1.44301i −0.692407 0.721507i \(-0.743450\pi\)
0.692407 0.721507i \(-0.256550\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.7949 1.04827 0.524135 0.851635i \(-0.324389\pi\)
0.524135 + 0.851635i \(0.324389\pi\)
\(864\) 0 0
\(865\) 2.60612 0.0886108
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 62.9253i − 2.13459i
\(870\) 0 0
\(871\) 18.7980i 0.636945i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.65685 0.191237
\(876\) 0 0
\(877\) 29.1010 0.982672 0.491336 0.870970i \(-0.336509\pi\)
0.491336 + 0.870970i \(0.336509\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.5371i 1.39942i 0.714427 + 0.699710i \(0.246688\pi\)
−0.714427 + 0.699710i \(0.753312\pi\)
\(882\) 0 0
\(883\) − 1.65153i − 0.0555784i −0.999614 0.0277892i \(-0.991153\pi\)
0.999614 0.0277892i \(-0.00884672\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.3991 0.550628 0.275314 0.961354i \(-0.411218\pi\)
0.275314 + 0.961354i \(0.411218\pi\)
\(888\) 0 0
\(889\) 10.8990 0.365540
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 50.3402i 1.68457i
\(894\) 0 0
\(895\) − 31.1918i − 1.04263i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 45.7155 1.52470
\(900\) 0 0
\(901\) 2.55051 0.0849698
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 27.2987i − 0.907438i
\(906\) 0 0
\(907\) − 41.1918i − 1.36775i −0.729598 0.683876i \(-0.760293\pi\)
0.729598 0.683876i \(-0.239707\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 19.2275 0.637037 0.318518 0.947917i \(-0.396815\pi\)
0.318518 + 0.947917i \(0.396815\pi\)
\(912\) 0 0
\(913\) −21.7980 −0.721407
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.36773i 0.0781892i
\(918\) 0 0
\(919\) − 21.5959i − 0.712384i −0.934413 0.356192i \(-0.884075\pi\)
0.934413 0.356192i \(-0.115925\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13.6814 −0.450330
\(924\) 0 0
\(925\) −2.69694 −0.0886748
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 56.9185i 1.86744i 0.358011 + 0.933718i \(0.383455\pi\)
−0.358011 + 0.933718i \(0.616545\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.65685 −0.184999
\(936\) 0 0
\(937\) 17.3031 0.565266 0.282633 0.959228i \(-0.408792\pi\)
0.282633 + 0.959228i \(0.408792\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.4203i 0.698281i 0.937070 + 0.349141i \(0.113526\pi\)
−0.937070 + 0.349141i \(0.886474\pi\)
\(942\) 0 0
\(943\) − 1.10102i − 0.0358542i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.38551 0.142510 0.0712549 0.997458i \(-0.477300\pi\)
0.0712549 + 0.997458i \(0.477300\pi\)
\(948\) 0 0
\(949\) −21.7980 −0.707592
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.5065i 0.437517i 0.975779 + 0.218759i \(0.0702007\pi\)
−0.975779 + 0.218759i \(0.929799\pi\)
\(954\) 0 0
\(955\) 31.1918i 1.00934i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −19.5133 −0.630116
\(960\) 0 0
\(961\) −46.4041 −1.49691
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 11.0280i − 0.355003i
\(966\) 0 0
\(967\) − 19.1010i − 0.614247i −0.951670 0.307124i \(-0.900633\pi\)
0.951670 0.307124i \(-0.0993665\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.38891 −0.237121 −0.118561 0.992947i \(-0.537828\pi\)
−0.118561 + 0.992947i \(0.537828\pi\)
\(972\) 0 0
\(973\) −15.7980 −0.506459
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.1278i 0.483980i 0.970279 + 0.241990i \(0.0778001\pi\)
−0.970279 + 0.241990i \(0.922200\pi\)
\(978\) 0 0
\(979\) − 67.3939i − 2.15392i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.09978 −0.130763 −0.0653813 0.997860i \(-0.520826\pi\)
−0.0653813 + 0.997860i \(0.520826\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1.73205i − 0.0550760i
\(990\) 0 0
\(991\) 46.4949i 1.47696i 0.674276 + 0.738480i \(0.264456\pi\)
−0.674276 + 0.738480i \(0.735544\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −58.8255 −1.86489
\(996\) 0 0
\(997\) 31.4495 0.996015 0.498008 0.867173i \(-0.334065\pi\)
0.498008 + 0.867173i \(0.334065\pi\)
\(998\) 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 6048.2.h.e.2591.6 yes 8
3.2 odd 2 inner 6048.2.h.e.2591.1 8
4.3 odd 2 inner 6048.2.h.e.2591.7 yes 8
12.11 even 2 inner 6048.2.h.e.2591.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.e.2591.1 8 3.2 odd 2 inner
6048.2.h.e.2591.4 yes 8 12.11 even 2 inner
6048.2.h.e.2591.6 yes 8 1.1 even 1 trivial
6048.2.h.e.2591.7 yes 8 4.3 odd 2 inner